Competition, Product Safety, and Product Liability 1

Transcription

1 Competition, Product Safety, and Product Liability 1 Yongmin Chen 2 University of Colorado Boulder and Zhejiang University Xinyu Hua 3 Hong Kong University of Science and Technology Abstract. A rm s incentive to invest in product safety is a ected by both market environment and product liability. We investigate the relationship between competition and product liability in a spatial model of oligopoly, where reputation provides a market incentive for safety investment and higher liability may distort consumers incentive for product care. We nd that partial liability, together with reputation concerns, can motivate rms to make safety investment. Increased competition due to less product di erentiation diminishes a rm s gain from maintaining reputation and raises the socially desired product liability. On the other hand, an increase in the number of competitors reduces the bene t from maintaining reputation, but has a non-monotonic e ect on the potential gain from cutting back safety investment; consequently, the optimal liability may vary non-monotonically with the number of competitors. In general, therefore, the relationship between competition and product liability is subtle, depending on how competition is measured. 1

2 1. INTRODUCTION Market competition and product liability are two major mechanisms that a ect rms incentives to increase product safety and prevent product harm to consumers. Extensive studies in law and economics have examined the e ects of product liability and derived optimal liability under a given market structure. However, there has been little formal analysis on how competition and product liability may interact to incentivize rms. 4 This is rather surprising, given the importance of product safety in many consumer markets. What is the relationship between competition and the socially desired liability level? Are competition and product liability substitutes or complements in increasing product safety and social welfare? This paper provides an economic analysis that aims to shed light on these questions. We consider a two-period spatial model with N 2 rms selling di erentiated products to heterogeneous consumers. The products may malfunction and cause consumer harm with some probability. At the beginning of the rst period, each rm can decide whether to invest in product safety. Investment leads to a safer product that causes less damage when it fails. Consumers cannot observe rms safety investments, but in the second period they can observe the damages to the harmed consumers in the rst period and update their beliefs about product safety. That is, there are reputation concerns for rms. If a rm s product causes consumer harm, it may bear product liability and compensate consumers. 2

3 In particular, under partial liability the rm is required to compensate only a proportion of consumer loss, whereas under full liability the rm is liable for the full damage. While liability can motivate rms to invest in product safety, it is not without undesirable incentive consequences due to the presence of two-sided moral hazard: After purchase, consumers can take precaution to reduce the potential harm from product failure; and higher product liability lowers the consumers precaution e orts. We are interested in situations where investment in product safety is socially desirable. The choice of product liability level is thus concerned with how to sustain the equilibrium where rms make safety investment. If a rm chooses a low-safety product, it faces high (expected) liability costs and reputation loss. When reputation loss is high enough, the rm will make the safety investment even without product liability, in which case the socially optimal liability is zero in order to encourage consumer precaution. However, if reputation concern is not su cient, then product liability needs to be increased in order to sustain the rm s investment incentive. For a given number of competitors in the market, less horizontal product di erentiation reduces equilibrium price, which diminishes the reputation loss in period 2 from o ering a low-safety product. Consequently, the socially optimal liability level is larger when competition becomes tougher due to reduced product di erentiation. In this sense, competition and product liability are complements in improving product safety and social welfare. However, the socially optimal liability level may vary non-monotonically with another measure of competition intensity: the number of competitors. If a rm deviates from a high-safety to a low-safety product, its gain in period 1 depends on the xed investment 3

4 cost saving and the change in the e ective marginal cost, which is the extra liability cost minus variable production cost. A rm would sell a high-safety product only when the reputation loss is larger than the deviation gain in period 1. In our model, by reducing equilibrium output from each rm, an increase in the number of competitors always reduces the reputation loss from deviation, but has a non-monotonic impact on the deviation gain in period 1, depending on the change in the e ective marginal cost. As a result, the optimal liability may vary non-monotonically with the number of competitors, possibly rst decreasing and then increasing. Thus, in general, the relationship between competition and product liability is subtle, depending on the measure of competition intensity. While they can often be complements, the relationship may also be non-monotonic when competition is measured by the number of competitors in the market. Our results can shed light on the mixed empirical evidence concerning the e ects of competition on rms investment incentives for product safety. For example, a 2008 survey among product development managers in the US revealed that companies were more likely to reduce safety investment and to speed up new product introductions, possibly with lower safety, when facing more competition (Samra et al., 2008). In the automobile industry, when more companies entered the market of SUVs, many products had low quality and later caused substantial consumer harm (Los Angeles Times, March 14, 2010). However, there have also been empirical studies showing that competition can increase product quality, though most of the studies do not focus on safety issues. 5 Furthermore, there is empirical evidence suggesting that product liability can have non-monotonic e ects on rms innovation incentives (e.g., Viscusi and Moore, 1993). 4

5 Our paper contributes to the literature on product liability. Studies with a single rm analyze, for example, the e ects of product liability on a rm s precaution to ensure product safety (Simon, 1981), or on both before-sales and after-sales precaution (Ben-Shahar, 1998; Chen and Hua, 2012), or on its incentive to disclose quality information through price and other devices (Daughety and Reinganum, 1995, 2008a). 6 Studies with competition include Epple and Raviv (1978), Polinsky and Rogerson (1983), Cooper and Ross (1984), and Daughety and Reinganum (2006, 2008b, 2014). Polinsky and Shavell (2010) argue that market mechanisms and product liability are substitutes as they both can increase product safety. Ganuza et al. (2016) study the roles of liability and reputation in a repeated game setting with one rm and one consumer. They show that the reputation mechanism can motivate the rm to take e ort that reduces the consumer s expected damage, but it causes social loss when the consumer does not buy in the punishment periods. Having liability can reduce such social loss. 7 Our study di ers from the literature by providing a formal analysis on the relationship between product liability and competition, and we nd that product liability and competition can be either complements or substitutes under alternative measures of competition. Our paper is also related to the literature in industrial organization, where market reputation can be an e ective mechanism to improve product quality (Shapiro, 1983; Allen, 1984; Bagwell and Riordan, 1991; Kranton, 2003), and where market competition may have either positive or negative impacts on product quality (Riordan, 1986; Shaked and Sutton, 1987; Horner, 2002; Dana and Fong, 2011). We depart from the literature by focusing on safety investments and product liability. 5

6 The rest of the paper is organized as follows. Section 2 presents our model and derives consumers precaution e ort in equilibrium. Section 3 analyzes rms investment incentives and how changes in competition, measured alternatively by product di erentiation and the number of competitors, a ect the socially optimal liability level. Section 4 discusses the robustness of our results and some alternative policies. Section 5 concludes. Proofs of the main results are gathered in the appendix. An online appendix contains the proofs for Lemma 3 and Lemma 5, as well as detailed analysis for the discussion in Section 4 under the alternative modelling assumptions THE MODEL A market has N 2 rms and a unit mass of consumers. Consumers are uniformly distributed on a network of N(N 1) 2 arcs of length 1, and the density of consumers on each 2 arc is thus N(N 1) : Each rm is uniquely located at one end of each of N 1 arcs. In the static form of the model, rms choose prices simultaneously, with rm i competing with every other rm j on a separate arc l ij ; for j 6= i and i; j = 1; :::; N: Each consumer, who values the product at V and demands at most one unit, must travel to a rm in order to make a purchase, with unit transportation cost t > 0: A consumer on l ij is uniquely denoted by x ij 2 [0; 1] ; whose distance is x ij from rm i and 1 x ij from rm j: Consumer x ij will purchase the product if her net surplus from the product V minus price and transportation cost is non-negative, and she patronizes the rm with the highest net surplus between the two rms at the two ends of l ij to which she belongs, i and j: Adapted from Chen and Riordan s (2007) spokes model, this model provides a tractable formulation 6

7 of oligopoly price competition that extends the Hotelling analysis to any number of rms with non-localized competition, where e ectively each rm competes with every rm else for di erent segments of consumers. 9 Notice that this formulation reduces to the standard Hotelling model when N = 2. Figure 1 illustrates the model in the cases of N = 3 and N = 4: Notice also that when another rm is added to an N- rm model, there are N more l ij arcs. Since the overall mass of consumers is held xed, they are reallocated across the arcs. In Figure 1, when N = 3; there are 3 arcs and 1/3 of the consumers are on each arc; if another rm is added, there are now N + 1 = 4 rms and 6 arcs, with each arc containing 1/6 of the consumers. <<COMP: Place Figure 1 about here>> The static model described above is then embedded into a simple two-period dynamic game with safety investment and product liability. Speci cally, each rm s product may cause consumer harm with probability. At the beginning of Period 1, a rm can choose to invest k > 0; which enables it to produce a high-safety product in both periods at marginal cost c 0: Without the investment, the product will have low safety and zero marginal cost. After purchasing a product, a consumer can take precaution e ort. Without such e ort, if a consumer is harmed, her damage is d from a high-safety product and D > d from a low-safety product. 10 We de ne z D d, and assume c < z. Then, when the xed cost of safety investment is su ciently small, it is e cient for rms to produce and sell the high-safety product. 7

8 With precaution e ort, a consumer can reduce the damage by 2 [0; d). 11 Each consumer s precaution cost is (), which is strictly increasing and convex, with (0) = 0; 0 (0) = 0; and 0 (d) > : With consumer precaution, the expected damage level from a high-safety product is (d ), and the expected damage level from a low-safety product is (D ). If a consumer is harmed, the rm is required to compensate the consumer fraction of the damage according to its product liability. The rm bears "partial liability" if < 1, "full liability" if = 1 and punitive damage compensation if > 1. For simplicity, we focus on scenarios with 1: 12 Liability can be interpreted in various ways: For example, under the strict liability rule, is the share of consumer damage to be born by the rm. Alternatively, can be interpreted as the probability that court nds the rm liable (and when found liable, the rm pays compensation equal to consumer damage), assuming that court cannot observe or infer the rm s investment decisions or consumers precaution levels. In neither period can consumers directly observe the rms investments or product safety. In period 2, however, rms and consumers observe the damage levels su ered by harmed consumers in period 1. Consequently, they learn about the safety level of each product. In particular, if product j causes damage D (or d) in period 1, then consumers and other rms in period 2 will believe that product j has low safety (or high safety). We denote rm j s total pro t in two periods as j (I j ; I j j B), where (I j ; I j ) is a vector of investments by rm j and the N 1 other rms, and B is consumers belief in period 1 about product safety. In our simple setting, as long as all rms have positive outputs in period 1, consumers in period 2 will always learn the true product quality, because with a continuum of consumers, 8

9 fraction of consumers will be harmed in period 1, and the damage levels su ered by them, which reveal product safety levels, are observed by all consumers at the beginning of period 2. If a rm has zero output in period 1 so that information about its past damage is not available, we assume that consumers and other rms in period 2 hold the same belief about that rm as consumers in period Hence beliefs by consumers in period 2 are not shown explicitly in j (I j ; I j j B): In period 1, consumer beliefs are consistent with rms equilibrium safety choices, and we discuss this further in the next section. To summarize, the timing of the model is as follows: Period 1: Stage 1: Each rm independently decides whether to invest in product safety, I j = 0 or k; j = 1; 2; :::; N. Each rm s product safety choice is its private information. Stage 2: Firms simultaneously and independently post their prices p j, j = 1; 2:::; N: Consumers, observing the prices (but not the safety choices), make possible purchases. Stage 3: Each consumer chooses her precaution e ort after purchase. State 4: If any consumer is harmed by a rm s product, the rm bears liability 1. Period 2: Consumers observe the damage levels to the harmed consumers. 14 Stages 2-4 in period 1 are then repeated, with rm j 0 s price in period 2 denoted by q j : 9

10 Throughout the paper, we make the following assumption to ensure the full coverage of the market: A1: V > 3 2t + c + [D + (d)]: Before analyzing rm strategy, we rst examine consumer precaution e ort and the ef- cient safety investment. If all N rms make the safety investment and all consumers purchase, in two periods the total costs for producing the high-safety product is Nk + 2c, with social bene t 2(D d) = 2z: Hence, without consumer precaution, it is e cient for all the rms to invest in safety if and only if 2z Nk + 2c. Regardless of whether the purchased product has high or low safety, each consumer will choose to minimize her expected loss (excluding liability compensation) from product malfunction: max f(1 ) ()g ; and the optimal () satis es the rst order condition: (1 ) 0 ( ) = 0: Since () is convex and 0 (d) > ; we have () < d for any ; and () decreases in. Intuitively, when product liability () is larger, consumers expect to receive more compensation from the rm if they are harmed, which reduces their incentive to take precaution. However, e ciency requires that every consumer takes precaution to maximize (): Therefore, as the result below states, given the rms investment decisions, the e ciency of consumer precaution e ort increases, or total welfare improves, when liability is smaller

11 Lemma 1 Given the rms safety investments, consumers precaution e ort () and total welfare are higher when product liability is lower. Lemma 1 will play an important role in our analysis to follow, where rms safety investment incentives will be shown to increase in product liability. The incentive con ict of product liability for consumers and rms will lead to some unique (possibly interior) liability that maximizes total welfare. We will be interested in situations where 2z N k 2c > maxf ()g; so that investment in high safety is always e cient. For this purpose our analysis will further assume A2: N N; with N 2z 2c maxf ()g k > 2: 3. COMPETITION AND PRODUCT LIABILITY In this section, we derive the rms equilibrium strategies and then address two questions. First, given the number of competitors, how will changes in the degree of horizontal product di erentiation, a measure of competition intensity, impact the socially optimal liability level? Second, how will the change in market structure (i.e., the number of competitors) a ect the optimal liability? In addressing these questions, we also investigate whether market competition and product liability are complements or substitutes in increasing product safety and welfare. 11

12 3.1 HIGH-SAFETY EQUILIBRIUM We focus on the symmetric equilibrium where all N rms invest in product safety (i.e., choose high safety) and charge the same price. To derive the equilibrium, we rst characterize the rms pricing decisions and pro ts in period 2, and then discuss consumer belief in period 1 and the rms strategies. For simplicity, we maintain the following assumption in the remaining analysis: A3: t > t 1 (z c) 2 : 16 In the symmetric high-safety equilibrium, each rm sells positive output in period 1. Then in period 2, consumers observe each rm s product safety, because with a continuum of consumers, a positive portion of consumers in period 1 have experienced product malfunction from each rm and the damage level reveals each rm s product safety. Along the equilibrium path in period 2, for any consumer x ij 2 [0; 1] on the arc l ij, i 6= j; and i; j 2 f1; 2; :::; Ng; she is indi erent between products i and j if V x ij t q i [(1 )(d )+( )] = V (1 x ij )t q j [(1 )(d )+( )]; (1) where [(1 )(d ) + ( )]; which we shall call a consumer s expected damage, consists of her expected loss when a product malfunctions and her cost of exerting precaution e ort. Given the same product safety for both rms, a consumer s expected damages from the two products cancel each other in (1). Hence, given that all other rms charge price q ; the per-period demand for product j is 12

13 X i6=j;i2f1;::ng provided that 2 N max n t 2 N (N 1) maxft q j + q ; 0g = 2 t 2t N max qj + q ; 0 ; 2t o qj +q 2t ; 0 < 1: At the high-safety equilibrium, rm j chooses price q j to maximize its pro t in period 2: max[q j c (d )] 2 t q j N max qj + q ; 0 ; 2t where (d ) is the rm s expected liability cost per unit of sales. It is straightforward to establish the following: Lemma 2 Suppose that all rms choose high safety, i.e., I = k: Then, in period 2 there is a unique symmetric equilibrium where each rm sets price q = t + c + (d units of output, and earns a pro t of t N. ), sells 1 N Each rm s e ective marginal cost in each period is ^c [c + (d )] ; and the equilibrium price in period 2 is t + ^c; same as in the standard Hotelling model. Intuitively, each rm s pro t decreases when there are more competitors or when there is less product di erentiation. Notice that liability does not a ect the rms pro ts on the equilibrium path, because at the symmetric equilibrium where all rms have the same product safety, consumers face the same expected damage from all rms, so that the liability level merely shifts the equilibrium price (q ) without a ecting the equilibrium markup (q ^c). Suppose that one rm, say rm 1, has deviated to I = 0: As long as rm 1 still has positive output in period 1, it will be known as the low-safety rm in period 2. If any rm has zero output in period 1, to be consistent, we assume that consumers and other rms 13

14 in period 2 hold the same belief about this rm s product safety as consumers in period 1. We shall rst analyze the case with all rms having positive outputs. After we characterize the consumer belief in period 1, we then discuss the other case with some rm having zero output. In the rst case, where all rms have positive outputs, suppose that the prices are q j ; j = 1; :::; N following rm 1 s deviation. Denote the demand for product 1 in period 2 as F 1 (q 1 ; :::; q N ) ; and demand for any other product j 6= 1 as F j (q 1 ; :::; q N ) : Notice that, for any rm j 6= 1; it competes for two types of consumers: consumers located on l 1j, and consumers located on l ij ; for any i 6= j and i 6= 1. Firm 1 chooses q 1 to maximize its pro t in Period 2: max q 1 [q 1 (D )]F 1 (q 1 ; :::; q N ) : And rm j 6= 1 chooses q j to maximize its pro t in Period 2: max q j [q j c (d )] F j (q 1 ; :::; q N ) : As we show in the online appendix, with rms choosing Nash equilibrium prices in the subgame of period 2, F 1 (q 1 ; :::; q N ) > 0 if t > N 1 2N 1 (z c), which holds for any N given assumption A3 (t > (z c) 2 ). 17 We then have: Lemma 3 Suppose that rm 1 has low safety while the other rms have high safety, and that all rms have positive outputs in period 1. Then, rm 1 s pro t in period 2, with rms choosing Nash equilibrium prices in this subgame, is (t t 2) 2 ; where t 2 t 2 (N) = N 1 2N 1 (z c). From Lemma 3, liability does not a ect pro ts in period 2 even when rm 1 has low 14

15 safety and the other rms have high safety. To see the intuition behind this, consider competition between rm 1 and rm j 6= 1 in period 2: On one hand, rm 1 and rm j face di erent expected liability costs (z), so that they may charge di erent prices in the subgame. On the other hand, in period 2, consumers have the correct belief about the di erence in safety between rm 1 and rm j, and correspondingly the di erence in their expected loss ((1 )z), which a ects demand levels for the two rms products. By a ecting di erences both in rms liability costs and in consumers expected losses, liability level () only impacts rms equilibrium prices in this subgame but does not in uence their markups or output levels. 18 From Lemmas 2 and 3, a deviating rm s "reputation loss" in period 2 can be de ned as (N) 1 N t (t t 2 (N)) 2 t : Now consider the game in period 1. An important issue for our analysis is to specify consumer belief about product safety in period 1, when consumers only observe rms prices. While our equilibrium concept, perfect Bayesian equilibrium, requires that consumer beliefs must be consistent with rm strategies along the equilibrium path, it does not constrain o -equilibrium beliefs. Thus, there are potentially di erent o -equilibrium beliefs that could support a symmetric equilibrium where all rms choose high safety and also the same equilibrium price in period 1. For example, one possibility is that in period 1, when seeing an o -equilibrium price from a rm, consumers continue to believe that the rm has produced a high-safety product. Such an o -equilibrium belief imposes the least punishment on a 15

16 deviating rm, and if no rm could pro tably deviate under such a belief, one might think that the equilibrium is robust in some sense. 19 However, it could be more natural to allow consumers to hold reasonable beliefs that interpret o -equilibrium prices as "signals" of product safety, even though our model is not a standard signaling game. For this purpose, we shall adopt the approach used by Dana (2001) in specifying consumer beliefs o the equilibrium path. In a framework where consumers only observe prices but not rms capacity choices, Dana (2001) focuses on consumer belief that a rm has chosen capacity optimally, given its deviating price and the other rms equilibrium prices. 20 Similarly, we assume that consumers will believe the rm to have chosen product safety optimally, given its deviating price and the other rms equilibrium prices. Speci cally, denote p e as the equilibrium price in period 1 in a symmetric high-safety equilibrium, and B(p j ; p e ) as the consumer belief about product j s safety when rm j charges p j while all other rms charge p e. If consumers believe that product j has high safety, then B(p j ; p e ) = d; otherwise, B(p j ; p e ) = D. Given any price p j 6= p e and the prior that all products have high safety; if choosing high safety can indeed bring rm j a total pro t (in two periods) no less than that from choosing low safety, then consumer belief should be B(p j ; p e ) = d: On the other hand, given p j 6= p e and the prior that all products have high safety; if choosing low safety can generate strictly higher pro t for rm j than choosing high safety, then consumer belief should be B(p j ; p e ) = D: 21 Notice that, given the prior that all products have high safety, rm j has positive output if p j < p e + t: For any p j p e t; rm j has the same output and therefore consumer belief satis es B(p j ; p e ) = B(p e t; p e ): For p j 2 [p e t; p e + t); we show in the appendix that 16

17 there exists some unique bp solving (N) = k (z c) such that consumer belief takes the following form: t bp + pe ; (2) (1) If z c = 0; then B(p j ; p e ) = d if (N) k while B(p j ; p e ) = D if (N) < k; (2) If z c < 0; then B(p j ; p e ) = d if p j bp while B(p j ; p e ) = D if p j < bp; (3) If z c > 0; then B(p j ; p e ) = d if p j bp while B(p j ; p e ) = D if p j > bp. This set of consumer beliefs has an intuitive interpretation. When deviation to low safety does not change rm j s e ective marginal cost (z c = 0); consumers believe that product j has high safety if the reputation loss (N) is larger than the xed cost of safety investment k, and low safety if otherwise. When deviation to low safety reduces rm j s e ective marginal cost (z c < 0); consumers hold the belief that product j has high safety if the rm charges a price higher than a certain threshold bp, and low safety if otherwise. Similarly, when deviation to low safety increases rm j s e ective marginal cost (z c > 0); consumers hold the belief that product j has high safety if the rm charges a price lower than bp, and low safety if otherwise. Finally, if p j p e + t, consumers would not buy product j no matter what their belief is, and hence rm j has zero output in period 1. It can also be veri ed that a high-safety rm would never prefer zero output to positive output. For convenience, we thus assume that, if p j p e + t; consumers in period 1 hold the belief that product j has low safety, and consistently, consumers and other rms in period 2 will hold the same belief that product 17

18 j has low safety. Let N be such that (N) = k N z c N : One can verify that bp 6= p e if > N ;while bp = p e if = N : Furthermore, if N and z c = 0, we have (N) k z c N = k; so that B(p j ; p e ) = d for any p j 2 [p e t; p e + t): The following lemma summarizes the above characterization of consumer beliefs and further shows that high-safety equilibrium does not exist if < N : Lemma 4 The high-safety equilibrium does not exist if < N : If N ; for p j p e + t; consumer belief in both periods satis es B(p j ; p e ) = D; for p j < p e t; consumer belief in period 1 satis es B(p j ; p e ) = B(p e t; p e ); and for p j 2 [p e t; p e + t), consumer belief takes the following form: (1) Suppose that z c = 0: Then B(p j ; p e ) = d. (2) Suppose that z c < 0: Then there exists some unique bp; as de ned in (2), such that B(p j ; p e ) = d if p j bp but B(p j ; p e ) = D if p j < bp; also bp p e ; with strict inequality if and only if > N : (3) Suppose that z c > 0: Then there exists some unique bp; as de ned in (2), such that B(p j ; p e ) = d if p j bp; but B(p j ; p e ) = D if p j > bp; also bp p e ; with strict inequality if and only if > N : Based on Lemma 4, when N and z c = 0, or > N and z c 6= 0; if rm j charges a price that is in a small neighbourhood of p e, consumers will hold the belief that product j has high safety. With a high-safety product, if rm j chooses a price much higher or lower than p e so that consumers would believe that it has deviated to low safety, then its pro t in period 1 would be lower. Thus, in any high-safety equilibrium in period 1, the optimal price for rm j solves the following problem: max[p j c (d )] t p j + p e : p j 18

19 We then have the optimal price for rm j satisfying p e = pe 2 + t + c + (d ) pe p 2 ; which immediately implies p e = p = t + c + (d ): Therefore, whenever N and z c = 0, or > N and z c 6= 0; if high-safety equilibrium exists, it must be unique with all rms charging p in period 1. However, when = N and z c 6= 0; without further re nement, we can have a continuum of high-safety equilibria. For example, under the above consumer belief, it can be veri ed that, with z c < 0; there can exist a continuum of equilibria with prices p e greater than or equal to p : 22 Similarly, when z c > 0; there may also exist a continuum of equilibria with prices lower than or equal to p : Notice that this multiplicity of highsafety equilibria only occurs when = N : But since the unique equilibrium price is p when > N ; one might consider the equilibrium with p e = p when = N as being more robust, in the sense that the equilibrium strategy when = N is stable with respect to a small perturbation of the environmental variable : We formalize this idea by considering the following stable equilibrium re nement when there are multiple equilibria: De nition 1 An equilibrium associated with a speci c liability level in our model is stable if, in period 1, its equilibrium price is the limit of equilibrium prices for liability levels that are larger than but arbitrarily close to : 19

20 This concept of stable equilibrium shares some similar motivation as the trembling-hand perfect equilibrium (Selten, 1975), even though they are not the same. 23 In what follows, we shall focus on the stable equilibrium when there exist multiple equilibria. Therefore, for any N ; the unique equilibrium price in period 1 is p e = p : Combining the above analysis for period 1 with that for period 2, we see that when the (stable) high-safety equilibrium exists, it is unique with all rms charging p = q in both periods and earning total pro ts 2t N k in two periods together: We next characterize the conditions for the existence of the equilibrium. Given Lemma 4, any high-safety equilibrium does not exist if < N : So suppose that N : To sustain the high-safety equilibrium, we need to consider two types of deviations by a rm: rst, deviating to low safety but charging a price leading consumers to believe that it has high safety; and second, deviating to low safety and charging a price leading consumers to believe that it has low safety. For the rst type of deviation, Lemma 4 implies that, in order for consumers in period 1 to believe that it has high safety, the deviating rm has to choose a price less than p + t and therefore has positive output, which reveals its true product safety to consumers and other rms in period 2. Then, as long as N ; based on our earlier analysis of consumer belief, the rm will have no incentive to deviate to low safety. Now consider the second type of deviation. At the proposed equilibrium where all rms choose high safety and charge p, if a rm, say rm 1; deviates to low safety and charges a price such that B(p 1 ; p ) = D; the demand for product 1 in period 1 would be 2 N maxft p 1 + p (1 )z ; 0g; 2t 20

21 provided t p 1+p (1 )z 2t < 1: Notice that, given B(p 1 ; p ) = D; the deviating rm s pro t in period 2 would be the same, no matter whether it has positive or zero output in period In period 1, ignoring the constraint B(p 1 ; p ) = D; rm 1 s optimal price after deviation therefore maximizes its period 1 pro t: maxf[p 1 (D )] 2 p 1 N maxft p 1 + p (1 )z ; 0gg: 2t The optimal deviating price in period 1 is ~p 1 = t+ (D ) z c 2 : Given A3 (t > t 1 = z c 2 ); if B(~p 1; p ) = D, rm 1 has positive output and its deviating pro t in period 1 is [~p 1 (D )] 2 N t z c 2 2t = (t t 1) 2 : Accordingly, if B(~p 1 ; p ) = D; the di erence between rm 1 s equilibrium pro t and deviation pro t in two periods is ( 2t N k) [(t t 1) 2 + (t t 2) 2 ] = [ t N (t t 1 ) 2 ] + (N) k: Hence, if [ t N (t t 1 ) 2 ] + (N) k; rm 1 would not deviate to low safety with a deviating price such that B(p 1 ; p ) = D. It follows that a su cient condition for the existence of the high-safety equilibrium is: N and [ t N (t t 1 ) 2 ] + (N) k: In the appendix, we show that this condition is also necessary. Thus, we have Proposition 1 There exists a unique equilibrium where all rms produce the high-safety 21

22 product and charge p = t + c + (d ) in both periods if and only if N and [ t N (t t 1 ) 2 ] + (N) k: (3) It can be veri ed that condition (3) is consistent with (A2): De ne the socially optimal liability as : We prove in the appendix the following corollary of Proposition 1: Corollary 1 There exist two cut-o values k 1 ; k 2, with 0 k 1 < k 2 and both independent of : If k k 2 ; the high-safety equilibrium exists, with = 0 when k k 1 and = N 2 (0; 1] satisfying z c N +(N) = k when k 1 < k k 2 : If k > k 2 ; = 0; and the high-safety equilibrium does not exist. Corollary 1 characterizes the socially optimal liability that ensures the existence of the high-safety equilibrium. We next examine how the optimal liability depends on competition, considering in turn two alternative measures of competition intensity: product di erentiation between rms and the number of rms in the market. 3.2 PRODUCT LIABILITY AND PRODUCT DIFFERENTIATION In our spatial model of oligopoly, consumers unit transportation cost (t); which indicates their preference heterogeneity or the degree of horizontal product di erentiation, is a natural measure of the intensity of competition. When t decreases, consumers are less heterogeneous, which reduces product di erentiation and lowers equilibrium market prices. The result below shows that the optimal liability generally increases when competition is more severe in the sense that t decreases. Proposition 2 Holding all other parameter values constant, there exist two cut-o values 22

23 t L < t H such that: (i) when t < t L or t t H, the socially optimal liability = 0, and (ii) when t L t < t H ; is positive and strictly decreases in t: Thus, product liability and market competition tend to be complements, when competition intensity is measured by the degree of product di erentiation. When there is less product di erentiation, rms compete more aggressively, resulting in lower pro ts in both periods and lower bene t from being known as a high-safety producer in period 2. Then, if a rm deviates to low safety, its "reputation loss" in period 2 would be smaller. Consequently, to sustain the high-safety equilibrium, product liability should be increased to raise the deviation cost. Furthermore, when product di erentiation is su ciently small (t < t L ), competition is so erce that a rm s reputation loss from deviation becomes too small to overcome the deviation gain. Therefore, under any liability level, the high-safety equilibrium does not exist. Then, the socially optimal liability should be zero in order to maximize consumers precaution incentives. On the other hand, when product di erentiation is su ciently large (t t H ), the reputation loss from deviation is high enough to maintain a rm s investment incentive. Then, the high-safety equilibrium exists under any liability level, and thus again the socially optimal liability is zero. 3.3 PRODUCT LIABILITY AND THE NUMBER OF COMPETITORS We next examine how the optimal liability may vary with the number of competitors. A change in the number of competitors a ects not only the reputation loss from deviation; but also each rm s equilibrium output level that has a non-monotonic impact on the deviation 23

24 gain in period 1. The net e ect of a change in the number of competitors on the optimal liability can thus be ambiguous. For convenience, we shall treat N as a continuous variable in our analysis. As shown in the online appendix, the following lemma states a monotonic relationship between the reputation loss from deviation and the number of competitors. Lemma 5 At the high-safety equilibrium, a rm s reputation loss from deviating to low safety, (N); strictly decreases in N: From Corollary 1, when the socially optimal liability is positive, it satis es (N) = k z N c ; where the term on the right-hand side is the potential gain from deviating to low safety but charging the equilibrium price in period 1, which includes saving of the xed investment cost and pro t variation due to the marginal cost change ( z c). If (N) k; then z c 0; that is, deviation reduces the e ective marginal cost; and if (N) < k; then z c > 0, implying higher e ective marginal cost from deviation: Therefore, this deviation gain in period 1 may decrease or increase in N; depending on the sign of z c: De ne (N) = d[n(n)] : dn In the appendix, we show that N(N) is a concave function in N. The result below demonstrates that, holding all other parameters constant, the optimal liability may vary non-monotonically with the number of competitors. Proposition 3 Suppose that > 0 for N 2 [N 1 ; N 2 ] ; with 2 N 1 < N 2 N: (i) If 24

25 (N 2 ) < k < (N 1 ); then there exists some N b 2 (N 1 ; N 2 ) such that for any N 2 [N 1 ; N 2 ], the optimal liability decreases in N for N < N b and increases in N for N > N: b (ii) If k (N 2 ); then decreases in N for any N 2 [N 1 ; N 2 ]: (iii) If k (N 1 ); then increases in N for any N 2 [N 1 ; N 2 ]: If a rm deviates to low safety and still charges the equilibrium price, it bene ts from saving the xed cost of investment and the variable production cost in period 1, but su ers from the extra liability cost in period 1 and the reputation loss in period 2. A rm would sell a high-safety product only when the reputation loss is larger than the deviation gain in period 1. An increase in the number of competitors always reduces the reputation loss from deviation (due to each rm s lower output), while the deviation gain in period 1 may vary non-monotonically with the number of competitors. Now, to see the intuition behind part (i) in Proposition 3, consider two cases when (N 2 ) < k < (N 1 ): First, if the number of competitors (N) is relatively small, reputation loss from deviation is large, so that the optimal extra liability cost to sustain the high-safety equilibrium is smaller than the variable production cost (i.e., z < c). In this case, deviation not only saves a rm s xed cost of investment, but also provides a reward of lower e ective marginal cost. A rm s total deviation gain in period 1 then decreases in N, as an increase in N reduces each rm s output level. Thus, when N increases, while the decreased reputation loss raises the rms incentive for deviation, the decreased deviation gain in period 1 reduces it. For small enough N, the latter e ect dominates, and hence the optimal liability decreases in N. In this case, competition and product liability are substitutes to achieve high product safety and also e ciency. However, for larger N, the former e ect dominates, and hence 25

26 the optimal liability increases in N. In this case, competition and product liability become complements to achieve high product safety and e ciency. Second, if the number of competitors is large enough; the reputation loss from deviation is small, so that the optimal extra liability cost sustaining the high-safety equilibrium is larger than the variable production cost. In this case, deviation saves a rm s xed cost, but generates a "penalty" of higher e ective marginal cost. Thus, a deviating rm s total gain in period 1 increases in the number of competitors. When N increases, both the decreased reputation loss and the increased deviation gain in period 1 raise the rms incentive for deviation. Therefore, the optimal liability must be increased to maintain the rms investment incentive. In this case, competition and product liability continue to be complements to achieve high product safety and e ciency. To see the intuition behind (ii) and (iii) of Proposition 3, notice that if k is small enough, then z is much smaller than c: Thus, a higher N signi cantly lowers a rm s deviation gain in period 1 and this e ect always dominates the reputation e ect. Hence is optimally set lower: However, if k is large enough, then to sustain the high-safety equilibrium, liability must be high enough such that deviation increases the e ective marginal cost. In this case, a higher N raises the rm s deviation incentive and hence needs to be higher: The numerical example below illustrates the possibility that may rst decrease and then increase in N as in Proposition 3. Example 1 Let z = 2; t = 1; c = 1 and k = 0:01: 25 In addition, let max f ()g < 1: Then, we nd that > 0 for any N 2 [2; N]; and the socially optimal liability rst decreases in N when N < b N = 6 and then increases in N when 6 < N < N; as shown in 26

27 Figure 2 below: <<COMP: Place Figure 2 about here>> The following corollary considers one special case of Proposition 3. When the variable cost of providing high safety becomes zero, the socially optimal liability always increases with the number of competitors. Corollary 2 If c = 0 and > 0 for N 2 [N 1 ; N 2 ] ; with 2 N 1 < N 2 N; then strictly increases in N for any N 2 [N 1 ; N 2 ]: The results in this section can lead to useful policy implications. For example, given the number of competitors, in a mature market with relatively small product di erentiation, product liability should be large enough to sustain rms incentives in making safety investments; in contrast, when the market features larger product di erentiation possibly due to new innovation or di erent technologies, as implied by Proposition 2, product liability can be reduced, or equivalently, courts can adopt a higher standard for evidence in deciding whether a rm is liable for consumer damage. This implication is aligned with the general idea that more protection (here less product liability) should be given to rms in markets with new innovation or di erent technologies, but in our paper, this result is based on the relationship between product liability and competition intensity as measured by product di erentiation. For another example, in choosing product liability, courts and regulators should consider the number of competitors as well as rms cost structure in making safety investment, although it may be di cult to set the optimal liability level based on the number of com- 27

28 petitors. In practice, rms can adopt various technologies or methods to improve product safety and reduce consumer damage. If rms can take R&D projects to improve product safety and warn consumers about the potential harm, then the relevant costs for rms are mainly xed costs instead of variable costs. In such cases, as shown in Corollary 2, product liability and competition as measured by the number of competitors are complements. In industries with more competitors or when new entrants arrive, liability should be increased to motivate rms R&D e ort. In contrast, if rms not only incur xed R&D costs but also add safety devices with variable costs to increase product safety, then product liability and competition can be either substitutes or complements, as shown in Proposition 3. In particular, if the number of competitors is small enough (for example, in a new market), it is likely that product liability and competition are substitutes. In such cases, liability can be reduced when a new rm enters the market. Notice that in our model product liability does not a ect the equilibrium pro t and therefore would not change a rm s entry decision. In more general settings where rms pro ts depend on liability, however, changing product liability can have an extra e ect on entry decisions. The relationship between product liability and endogenous entry decisions can be an interesting topic for future research. 4. DISCUSSION We have conducted our analysis in a variant of the spokes model that extends the classic Hotelling duopoly. We wish to allow product di erentiation between rms in the market, for which the Hotelling model is known to have very desirable features. To extend Hotelling 28

29 to an oligopoly with any number of rms, one motivation to use the spokes model instead of, say, the circle model (Salop, 1979), is that when only one rm deviates from safety investment, in the subgame of period 2 there is naturally a two-price equilibrium that is easy to characterize in the spokes model, because all N 1 rms remain symmetric to each other and with respect to the deviating rm. By contrast, in the circle model, rms are not symmetric as they are located farther away from the deviating rm in each direction, and hence in the equilibrium of period 2 following a rm s deviation there could be more than N 2 distinctive prices, which would be extremely di cult to characterize for an arbitrary N: 26 While our spatial model is relatively simple and intuitive to work with, it imposes speci c structures that can be restrictive. 27 We have also made other simplifying assumptions in the model to make the analysis tractable. In this section, we rst discuss several alternative formulations to shed light on the robustness of our results. Speci cally, we consider in turn an alternative demand system, an alternative formulation on consumer precaution e orts, and an alternative assumption on observability in period 2. Our main insights concerning the relationship between competition and product liability remain valid in these alternative settings. Speci cally, competition and product liability can be either complements or substitutes in their roles to promote product safety, and the nature of the relationship may depend on how competition is measured. Furthermore, the relationship may be non-monotonic. An online appendix provides the detailed analysis corresponding to the rst three subsections below. At the end of this section, we also discuss some alternative liability rules. 29

30 4.1 ALTERNATIVE CONSUMER DEMAND WITH VARIABLE TOTAL OUT- PUT The demand in our main model, as in other spatial models of competition, has the restrictive feature that in equilibrium the market is fully covered with a xed total output. This removes the extensive margin that would also need to be taken into account in determining the socially optimal product liability. We now brie y explain that our main results concerning competition and product liability can also hold in an alternative model of consumer demand with variable total output (Daughety and Reinganum, 2008b), which is derived from the qausi-linear quadratic utility function of a representative consumer. Following Daughety and Reinganum (2008b), consider a single consumer who consumes Q j of product j, j = 1; :::N; with utility function U(Q 1 ; ::Q N ) = X j [A (1 )(B j j ) ( j )]Q j 1 2 (X j Q 2 j + X j X Q j Q i ); i6=j where B j = fd; dg is the consumer s belief about damage level caused by product j and > 0 is the degree of product substitution between any two products. Assume that > and A is large enough such that there is always positive output for each product. This utility function leads to the following demand for any product j Q j (p 1 ; ::p N ; B 1 ; :::B N ) = A 0 b[(1 )(B j j ) + ( j )] + g X [(1 )(B i j ) + ( j )] bp j + g X p i ; i6=j i6=j where A 0 = A [+(N 2)] +(N 1) ; b = ( )[+(N 1)], and g = ( )[+(N 1)] : 30

31 The full- edged analysis based on the general demand function becomes rather complicated. For simplicity, we consider two special scenarios investigating the relationship between liability and the di erent measures of competition, N and, respectively. First, under the assumption that = 2 and = 1; we examine the relationship between the optimal liability and the number of competitors (N). At the high-safety equilibrium, if a rm deviates to low safety, it gains pro t in period 1 while su ers from reputation loss in period 2. The deviation gain in period 1 decreases in, but varies non-monotonically in N; depending on the change in the e ective marginal cost (z c): In contrast, the reputation loss is independent of liability and decreasing in N: Notice that these e ects are the same as those in our main model. Following similar analysis in Section 3, we can show that the optimal liability, whenever positive, changes non-monotonically when N increases, possibly rst decreasing and then increasing. Second, under the assumption that N = 2; = 1;and 2 (0; 1); we explore the relationship between the optimal liability and the degree of product substitution (): While we are unable to derive analytically the general relationship; in the numerical examples we have analyzed, we nd that whenever the optimal liability > 0; it increases in : That is, when competition is more erce due to more product substitution, or equivalently, less product di erentiation, the optimal liability increases to maintain the rms investment incentives, same as in our main model. Therefore, as in our main model, the relationship between competition and the optimal liability is subtle, depending on how competition is measured. 31

32 4.2 CONSUMER PRECAUTION CONTINGENT ON PRODUCT SAFETY In our model, consumer precaution is assumed to reduce the harm of a faulty product by a certain amount (), and the precaution e ort is independent of the product safety level. An alternative assumption could be that consumer precaution can reduce the damage by a fraction. Then, in equilibrium consumer precaution e ort will depend on their belief about product safety. Nevertheless, our main results still hold: when competition increases due to less product di erentiation, the optimal liability tends to become higher; whereas when competition increases due to a larger number of competitors, the optimal liability can vary non-monotonically. Suppose that a consumer can reduce the damage by a fraction of 1 2 [0; 1). This is equivalent to assuming that a consumer can reduce the probability of harm to : Note that lower implies less expected damage. Each consumer s precaution cost is '(), which is strictly decreasing and convex, with ' 0 (0) = 1: A consumer s optimal precautions, when she believes that the product has high or low safety, respectively are (d; ) or (D; ) and solve: max f (1 )d '()g ; max f (1 )D '()g : Clearly, (D; ) 6= (d; ): That is, consumer precaution depends on her belief about product safety. To simplify notations, de ne the sum of consumer damage and precaution cost as (D; a) (D; )D + '((D; )) and (d; a) (d; )d + '((d; )): Similar to the analysis in Section 3, it can be shown that, whenever the optimal liability sustaining the 32